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Mathematics
Parabola
Previous Years Questions

Let $$\mathrm{PQ}$$ be a focal chord of the parabola $$y^{2}=36 x$$ of length 100 , making an acute angle with the positive $$x$$-axis. Let the ordina...
Let $$\mathrm{A}(0,1), \mathrm{B}(1,1)$$ and $$\mathrm{C}(1,0)$$ be the mid-points of the sides of a triangle with incentre at the point $$\mathrm{D}... Let$$R$$be the focus of the parabola$$y^{2}=20 x$$and the line$$y=m x+c$$intersect the parabola at two points$$P$$and$$Q$$. Let the point$$...
Let $$\mathrm{y}=f(x)$$ represent a parabola with focus $$\left(-\frac{1}{2}, 0\right)$$ and directrix $$y=-\frac{1}{2}$$. Then $$S=\left\{x \in \mat... Let A be a point on the x-axis. Common tangents are drawn from A to the curves x^2+y^2=8 and y^2=16 x. If one of these tangents touches the ... The parabolas : a x^2+2 b x+c y=0 and d x^2+2 e x+f y=0 intersect on the line y=1. If a, b, c, d, e, f are positive real numbers and a, b, c... If$$\mathrm{P}(\mathrm{h}, \mathrm{k})$$be a point on the parabola$$x=4 y^{2}$$, which is nearest to the point$$\mathrm{Q}(0,33)$$, then the dista... If the tangent at a point P on the parabola$$y^2=3x$$is parallel to the line$$x+2y=1$$and the tangents at the points Q and R on the ellipse$$\fra...
The equations of two sides of a variable triangle are $$x=0$$ and $$y=3$$, and its third side is a tangent to the parabola $$y^2=6x$$. The locus of it...
The distance of the point $$(6,-2\sqrt2)$$ from the common tangent $$\mathrm{y=mx+c,m > 0}$$, of the curves $$x=2y^2$$ and $$x=1+y^2$$ is :
The equations of the sides AB and AC of a triangle ABC are $$(\lambda+1)x+\lambda y=4$$ and $$\lambda x+(1-\lambda)y+\lambda=0$$ respectively. Its ver...
Let a tangent to the curve $$\mathrm{y^2=24x}$$ meet the curve $$xy = 2$$ at the points A and B. Then the mid points of such line segments AB lie on a...
Let the focal chord of the parabola $$\mathrm{P}: y^{2}=4 x$$ along the line $$\mathrm{L}: y=\mathrm{m} x+\mathrm{c}, \mathrm{m}>0$$ meet the parabola...
If the tangents drawn at the points $$\mathrm{P}$$ and $$\mathrm{Q}$$ on the parabola $$y^{2}=2 x-3$$ intersect at the point $$R(0,1)$$, then the orth...
If the length of the latus rectum of a parabola, whose focus is $$(a, a)$$ and the tangent at its vertex is $$x+y=a$$, is 16, then $$|a|$$ is equal to...
Let $$P(a, b)$$ be a point on the parabola $$y^{2}=8 x$$ such that the tangent at $$P$$ passes through the centre of the circle $$x^{2}+y^{2}-10 x-14 ... Let$$\mathrm{P}$$and$$\mathrm{Q}$$be any points on the curves$$(x-1)^{2}+(y+1)^{2}=1$$and$$y=x^{2}$$, respectively. The distance between$$P$$... The equation of a common tangent to the parabolas$$y=x^{2}$$and$$y=-(x-2)^{2}$$is The tangents at the points$$A(1,3)$$and$$B(1,-1)$$on the parabola$$y^{2}-2 x-2 y=1$$meet at the point$$P$$. Then the area (in unit$${ }^{2}$$... Let P : y2 = 4ax, a > 0 be a parabola with focus S. Let the tangents to the parabola P make an angle of$${\pi \over 4}$$with the line y = 3x + 5 to... Let PQ be a focal chord of the parabola y2 = 4x such that it subtends an angle of$${\pi \over 2}$$at the point (3, 0). Let the line segment PQ be a... If vertex of a parabola is (2,$$-$$1) and the equation of its directrix is 4x$$-$$3y = 21, then the length of its latus rectum is : If the equation of the parabola, whose vertex is at (5, 4) and the directrix is$$3x + y - 29 = 0$$, is$${x^2} + a{y^2} + bxy + cx + dy + k = 0$$, th... Let the normal at the point on the parabola y2 = 6x pass through the point (5,$$-$$8). If the tangent at P to the parabola intersects its directrix a... If the line$$y = 4 + kx,\,k > 0$$, is the tangent to the parabola$$y = x - {x^2}$$at the point P and V is the vertex of the parabola, then the slop... If$$y = {m_1}x + {c_1}$$and$$y = {m_2}x + {c_2}$$,$${m_1} \ne {m_2}$$are two common tangents of circle$${x^2} + {y^2} = 2$$and parabola y2 = x,... Let$$x = 2t$$,$$y = {{{t^2}} \over 3}$$be a conic. Let S be the focus and B be the point on the axis of the conic such that$$SA \bot BA$$, where A... A particle is moving in the xy-plane along a curve C passing through the point (3, 3). The tangent to the curve C at the point P meets the x-axis at Q... Let x2 + y2 + Ax + By + C = 0 be a circle passing through (0, 6) and touching the parabola y = x2 at (2, 4). Then A + C is equal to ___________.... Consider the parabola with vertex$$\left( {{1 \over 2},{3 \over 4}} \right)$$and the directrix$$y = {1 \over 2}$$. Let P be the point where the par... The length of the latus rectum of a parabola, whose vertex and focus are on the positive x-axis at a distance R and S (> R) respectively from the o... If two tangents drawn from a point P to the parabola y2 = 16(x$$-$$3) are at right angles, then the locus of point P is : A tangent and a normal are drawn at the point P(2,$$-$$4) on the parabola y2 = 8x, which meet the directrix of the parabola at the points A and B res... Let a parabola b be such that its vertex and focus lie on the positive x-axis at a distance 2 and 4 units from the origin, respectively. If tangents a... Let P be a variable point on the parabola$$y = 4{x^2} + 1$$. Then, the locus of the mid-point of the point P and the foot of the perpendicular drawn ... Let the tangent to the parabola S : y2 = 2x at the point P(2, 2) meet the x-axis at Q and normal at it meet the parabola S at the point R. Then the ar... Let L be a tangent line to the parabola y2 = 4x$$-$$20 at (6, 2). If L is also a tangent to the ellipse$${{{x^2}} \over 2} + {{{y^2}} \over b} = 1\$...
Let C be the locus of the mirror image of a point on the parabola y2 = 4x with respect to the line y = x. Then the equation of tangent to C at P(2, 1)...
If the three normals drawn to the parabola, y2 = 2x pass through the point (a, 0) a $$\ne$$ 0, then 'a' must be greater than :
The shortest distance between the line x $$-$$ y = 1 and the curve x2 = 2y is :
A tangent is drawn to the parabola y2 = 6x which is perpendicular to the line 2x + y = 1. Which of the following points does NOT lie on it?
If P is a point on the parabola y = x2 + 4 which is closest to the straight line y = 4x $$-$$ 1, then the co-ordinates of P are :
The locus of the mid-point of the line segment joining the focus of the parabola y2 = 4ax to a moving point of the parabola, is another parabola whose...
The centre of the circle passing through the point (0, 1) and touching the parabola y = x2 at the point (2, 4) is :
Let L1 be a tangent to the parabola y2 = 4(x + 1) and L2 be a tangent to the parabola y2 = 8(x + 2) such that L1 and L2 intersect at right angles....
If the common tangent to the parabolas, y2 = 4x and x2 = 4y also touches the circle, x2 + y2 = c2, then c is equal to :...
Let the latus ractum of the parabola y2 = 4x be the common chord to the circles C1 and C2 each of them having radius 2$$\sqrt 5$$. Then, the distan...
Let P be a point on the parabola, y2 = 12x and N be the foot of the perpendicular drawn from P on the axis of the parabola. A line is now drawn throu...
The area (in sq. units) of an equilateral triangle inscribed in the parabola y2 = 8x, with one of its vertices on the vertex of this parabola, is :...
If one end of a focal chord AB of the parabola y2 = 8x is at $$A\left( {{1 \over 2}, - 2} \right)$$, then the equation of the tangent to it at B is :...
The locus of a point which divides the line segment joining the point (0, –1) and a point on the parabola, x2 = 4y, internally in the ratio 1 : 2, is ...
If y = mx + 4 is a tangent to both the parabolas, y2 = 4x and x2 = 2by, then b is equal to :
The equation of common tangent to the curves y2 = 16x and xy = –4, is :
The tangents to the curve y = (x – 2)2 – 1 at its points of intersection with the line x – y = 3, intersect at the point :
Let P be the point of intersection of the common tangents to the parabola y2 = 12x and the hyperbola 8x2 – y2 = 8. If S and S' denote the foci of th...
If the line ax + y = c, touches both the curves x2 + y2 = 1 and y2 = 4$$\sqrt 2$$x , then |c| is equal to :
The area (in sq. units) of the smaller of the two circles that touch the parabola, y2 = 4x at the point (1, 2) and the x-axis is :-
If one end of a focal chord of the parabola, y2 = 16x is at (1, 4), then the length of this focal chord is :
The tangent to the parabola y2 = 4x at the point where it intersects the circle x2 + y2 = 5 in the first quadrant, passes through the point : ...
The shortest distance between the line y = x and the curve y2 = x – 2 is :
The equation of a tangent to the parabola, x2 = 8y, which makes an angle $$\theta$$ with the positive directions of x-axis, is :
The maximum area (in sq. units) of a rectangle having its base on the x-axis and its other two vertices on the parabola, y = 12 – x2 such that the re...
Let P(4, –4) and Q(9, 6) be two points on the parabola, y2 = 4x and let x be any point on the arc POQ of this parabola, where O is the vertex of this ...
If the area of the triangle whose one vertex is at the vertex of the parabola, y2 + 4(x – a2) = 0 and the othertwo vertices are the points of intersec...
The length of the chord of the parabola x2 $$=$$ 4y having equation x – $$\sqrt 2 y + 4\sqrt 2 = 0$$  is -
If the parabolas y2 = 4b(x – c) and y2 = 8ax have a common normal, then which on of the following is a valid choice for the ordered triad (a, b, c)?...
Let A(4, $$-$$ 4) and B(9, 6) be points on the parabola, y2 = 4x. Let C be chosen on the arc AOB of the parabola, where O is the origin, such that the...
Axis of a parabola lies along x-axis. If its vertex and focus are at distances 2 and 4 respectively from the origin, on the positive x-axis then which...
Equation of a common tangent to the circle, x2 + y2 – 6x = 0 and the parabola, y2 = 4x is :
If $$\theta$$ denotes the acute angle between the curves, y = 10 – x2 and y = 2 + x2 at a point of their intersection, the |tan $$\theta$$| is equa...
Let P be a point on the parabola, x2 = 4y. If the distance of P from the center of the circle, x2 + y2 + 6x + 8 = 0 is minimum, then the equation of ...
Tangent and normal are drawn at P(16, 16) on the parabola y2 = 16x, which intersect the axis of the parabola at A and B, respectively. If C is the cen...
Tangents drawn from the point ($$-$$8, 0) to the parabola y2 = 8x touch the parabola at $$P$$ and $$Q.$$ If F is the focus of the parabola, then the a...
Two parabolas with a common vertex and with axes along x-axis and $$y$$-axis, respectively intersect each other in the first quadrant. If the length o...
If y = mx + c is the normal at a point on the parabola y2 = 8x whose focal distance is 8 units, then $$\left| c \right|$$ is equal to :
If the common tangents to the parabola, x2 = 4y and the circle, x2 + y2 = 4 intersect at the point P, then the distance of P from the origin, is : ...
P and Q are two distinct points on the parabola, y2 = 4x, with parameters t and t1 respectively. If the normal at P passes through Q, then the minimum...
Let $$P$$ be the point on the parabola, $${{y^2} = 8x}$$ which is at a minimum distance from the centre $$C$$ of the circle, $${x^2} + {\left( {y + 6}... Let$$O$$be the vertex and$$Q$$be any point on the parabola,$${{x^2} = 8y}$$. If the point$$P$$divides the line segment$$OQ$$internally in the... The slope of the line touching both the parabolas$${y^2} = 4x$$and$${x^2} = - 32y$$is Given : A circle,$$2{x^2} + 2{y^2} = 5$$and a parabola,$${y^2} = 4\sqrt 5 x$$. Statement-1 : An equation of a common tangent to these curves is$$...
If two tangents drawn from a point $$P$$ to the parabola $${y^2} = 4x$$ are at right angles, then the locus of $$P$$ is
A parabola has the origin as its focus and the line $$x=2$$ as the directrix. Then the vertex of the parabola is at :
The equation of a tangent to the parabola $${y^2} = 8x$$ is $$y=x+2$$. The point on this line from which the other tangent to the parabola is perpendi...
The locus of the vertices of the family of parabolas $$y = {{{a^3}{x^2}} \over 3} + {{{a^2}x} \over 2} - 2a$$ is :
Let $$P$$ be the point $$(1, 0)$$ and $$Q$$ a point on the parabola $${y^2} = 8x$$. The locus of mid point of $$PQ$$ is :
If $$a \ne 0$$ and the line $$2bx+3cy+4d=0$$ passes through the points of intersection of the parabolas $${y^2} = 4ax$$ and $${x^2} = 4ay$$, then :
The normal at the point$$\left( {bt_1^2,2b{t_1}} \right)$$ on a parabola meets the parabola again in the point $$\left( {bt_2^2,2b{t_2}} \right)$$, th...
Two common tangents to the circle $${x^2} + {y^2} = 2{a^2}$$ and parabola $${y^2} = 8ax$$ are :

## Numerical

Let the tangent to the parabola $$\mathrm{y}^{2}=12 \mathrm{x}$$ at the point $$(3, \alpha)$$ be perpendicular to the line $$2 x+2 y=3$$. Then the squ...
Let a common tangent to the curves $${y^2} = 4x$$ and $${(x - 4)^2} + {y^2} = 16$$ touch the curves at the points P and Q. Then $${(PQ)^2}$$ is equal ...
The ordinates of the points P and $$\mathrm{Q}$$ on the parabola with focus $$(3,0)$$ and directrix $$x=-3$$ are in the ratio $$3: 1$$. If $$\mathrm{R... Let the tangent to the curve$$x^{2}+2 x-4 y+9=0$$at the point$$\mathrm{P}(1,3)$$on it meet the$$y$$-axis at$$\mathrm{A}$$. Let the line passing ... If the$$x$$-intercept of a focal chord of the parabola$$y^{2}=8x+4y+4$$is 3, then the length of this chord is equal to ____________. Let \mathrm{S} be the set of all \mathrm{a} \in \mathrm{N} such that the area of the triangle formed by the tangent at the point \mathrm{P}(\math... A triangle is formed by the tangents at the point (2, 2) on the curves$$y^2=2x$$and$$x^2+y^2=4x$$, and the line$$x+y+2=0$$. If$$r$$is the radius... Two tangent lines$$l_{1}$$and$$l_{2}$$are drawn from the point$$(2,0)$$to the parabola$$2 \mathrm{y}^{2}=-x$$. If the lines$$l_{1}$$and$$l_{...
The sum of diameters of the circles that touch (i) the parabola $$75 x^{2}=64(5 y-3)$$ at the point $$\left(\frac{8}{5}, \frac{6}{5}\right)$$ and (ii)...
Let PQ be a focal chord of length 6.25 units of the parabola y2 = 4x. If O is the vertex of the parabola, then 10 times the area (in sq. units) of $$\... A circle of radius 2 unit passes through the vertex and the focus of the parabola y2 = 2x and touches the parabola$$y = {\left( {x - {1 \over 4}} \ri...
Let the common tangents to the curves $$4({x^2} + {y^2}) = 9$$ and $${y^2} = 4x$$ intersect at the point Q. Let an ellipse, centered at the origin O, ...
Let P1 be a parabola with vertex (3, 2) and focus (4, 4) and P2 be its mirror image with respect to the line x + 2y = 6. Then the directrix of P2 is x...
A tangent line L is drawn at the point (2, $$-$$4) on the parabola y2 = 8x. If the line L is also tangent to the circle x2 + y2 = a, then 'a' is equa...
If the point on the curve y2 = 6x, nearest to the point $$\left( {3,{3 \over 2}} \right)$$ is ($$\alpha$$, $$\beta$$), then 2($$\alpha$$ + $$\beta$$) ...
Let y = mx + c, m > 0 be the focal chord of y2 = $$-$$ 64x, which is tangent to (x + 10)2 + y2 = 4. Then, the value of 4$$\sqrt 2$$ (m + c) is equ...
A line is a common tangent to the circle (x $$-$$ 3)2 + y2 = 9 and the parabola y2 = 4x. If the two points of contact (a, b) and (c, d) are distinct a...
If the tangent to the curve, y = ex at a point (c, ec) and the normal to the parabola, y2 = 4x at the point (1, 2) intersect at the same point on the...
Let a line y = mx (m > 0) intersect the parabola, y2 = x at a point P, other than the origin. Let the tangent to it at P meet the x-axis at the poi...
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